Normalized defining polynomial
\( x^{20} - x^{19} - 5 x^{18} + 6 x^{17} + 5 x^{16} - 7 x^{15} + 7 x^{14} - 14 x^{13} - 5 x^{12} + 14 x^{11} + 8 x^{10} + 14 x^{9} - 29 x^{8} - 18 x^{7} + 23 x^{6} + 5 x^{5} - x^{4} + x^{3} - 5 x^{2} - x + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(23382101649260778427129=7^{8}\cdot 43^{2}\cdot 1217^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $13.14$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $7, 43, 1217$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{7} a^{18} - \frac{1}{7} a^{16} - \frac{2}{7} a^{15} - \frac{1}{7} a^{14} - \frac{2}{7} a^{13} + \frac{1}{7} a^{12} - \frac{1}{7} a^{10} - \frac{1}{7} a^{9} + \frac{3}{7} a^{8} - \frac{1}{7} a^{7} + \frac{3}{7} a^{6} + \frac{2}{7} a^{5} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{70727587} a^{19} + \frac{1458966}{70727587} a^{18} - \frac{14002353}{70727587} a^{17} - \frac{2102895}{10103941} a^{16} + \frac{30718418}{70727587} a^{15} + \frac{2345027}{10103941} a^{14} - \frac{10847356}{70727587} a^{13} + \frac{30010090}{70727587} a^{12} - \frac{20092269}{70727587} a^{11} + \frac{10464672}{70727587} a^{10} - \frac{19661805}{70727587} a^{9} + \frac{4683215}{10103941} a^{8} - \frac{28381754}{70727587} a^{7} - \frac{31066172}{70727587} a^{6} - \frac{28835858}{70727587} a^{5} + \frac{12114358}{70727587} a^{4} + \frac{18610584}{70727587} a^{3} - \frac{1807102}{70727587} a^{2} + \frac{4326406}{10103941} a - \frac{19642374}{70727587}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 1163.45280108 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 61440 |
| The 126 conjugacy class representatives for t20n669 are not computed |
| Character table for t20n669 is not computed |
Intermediate fields
| 5.3.8519.1, 10.2.3556094689.1, 10.4.152912071627.1, 10.4.3120654523.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $7$ | 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.8.6.2 | $x^{8} - 49 x^{4} + 3969$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
| $43$ | 43.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 43.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.4.2.1 | $x^{4} + 215 x^{2} + 16641$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 43.4.0.1 | $x^{4} - x + 20$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 43.4.0.1 | $x^{4} - x + 20$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 1217 | Data not computed | ||||||